Design of hybrid systems with guaranteed performance

In this paper we show that, for a linear system, any worst-case energy gain greater than the optimal H_∞ norm is achievable by a logarithmically quantized state feedback. We also show how to derive the coarsest logarithmic quantizer provable via quadratic Lyapunov functions for a given level of performance. The smallest logarithmic base, for a given performance level, is obtained via a bisection algorithm applied to a parametric feasibility LMI problem. The result highlights the tradeoff between performance degradation versus coarseness of quantization. Simulations suggest that the upper bound derived in this paper is a realistic measure of the actual performance under logarithmic quantization. The end result is the systematic design of a discrete event controller that stabilizes a linear system and guarantees a certain level of performance measured in terms of the worst-case close loop energy gain. The resulting hybrid system is implicity verified.